Question

∫\(\frac {5(x^6+1)}{X+1}\)dx = (where C is a constant of integration.)

• A. \(\frac {5x^7}{7}\)+ 5x + 5 tan^-1 x + c
• B. 5 tan^–1 x + log (x² + 1) + C
• C. 5(x + 1) + log (x + 1) + C
• D. x⁵ – \(\frac {5x^3}{3}\) + 5x + C

Answer 1

The Correct Option is D

Solution:

Step 1: Let I = ∫ [5x(x⁴ + 1) / (x² + 1)] dx

Step 2: Rewrite the integrand by multiplying numerators:

I = 5 ∫ [x(x⁴ + 1)] / (x² + 1) dx = 5 ∫ [(x² + 1)(x⁴ − x² + 1)] / (x² + 1) dx

Here, the numerator is factored cleverly as (x² + 1)(x⁴ − x² + 1), so the (x² + 1) cancels out:

I = 5 ∫ (x⁴ − x² + 1) dx

Step 3: Break the integral:

I = 5 ∫ x⁴ dx − 5 ∫ x² dx + 5 ∫ 1 dx

Step 4: Integrate each term:

• ∫ x⁴ dx = x⁵ / 5
• ∫ x² dx = x³ / 3
• ∫ 1 dx = x

So,

I = 5 [x⁵ / 5 − x³ / 3 + x] + C

Simplify:

I = x⁵ − (5x³ / 3) + 5x + C

Final Answer: x⁵ − (5x² / 3) + 5x + C

Therefore, the correct option is: (D) x⁵ − (5x² / 3) + 5x + C